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Abstract
We propose a compact dual wavelength digital holographic Microscopy (DHM) based on a long working distance objective, which enabling quantitative phase imaging of opaque samples with extended measurement range in one shot. The compactness of the configuration is achieved by constructing a miniature modified Michelson interferometer between the objective and the sample, and as a result it provides higher temporal stability than conventional dual wavelength DHM. In the setup, the propagation directions of two reference beams of different wavelengths can be independently adjusted, and thus two off axis interferograms having orthogonal fringe directions can be simultaneously captured through a monochrome CCD camera. The unambiguous vertical measurement range in optical path length is extended to 8.338 μm, the length of a synthetic wavelength, by selecting two wavelengths with a gap of 52 nm. The capability of the proposed setup is demonstrated with measurements of a standard 1.8 μm height step as well as a moving micro staircase structure.
© 2017 Optical Society of America
1. Introduction
Digital holographic microscopy (DHM) is a powerful measuring technique which provides quantitative phase information related to microscopic specimens [1–6]. Due to its noncontact, wide-field and high precision nature, it has been widely used in surface micro-topography [7–11] and biological cell imaging [12–14]. In single-wavelength DHM, usually the wrapped phase within the range [-π π] is firstly constructed from a interferogram [3], and then a phase unwrapping algorithm is applied to unwrap it to get continuous phase map [15], which is proportional to the optical path lengths (OPL) of the tested specimen. In addition to the drawback that the unwrapping algorithm is usually computationally demanding, the maximum measurable OPL between adjacent sampling points of a specimen is restricted to one time of wavelength, specifically, the maximum height difference is half the wavelength in reflection mode when measuring abrupt steps; otherwise the algorithm will fail and result in an error.
To solve these problems, dual-wavelength interferometry or DHM is introduced [16–20]. By careful selecting two wavelengths, a longer synthetic wavelength can be obtained to extend the measurement range greatly. In Refs [16–20], multiple interferograms at different wavelengths are successively recorded. Since it is time-consuming and hence, these methods are incapable of measuring moving object or dynamic phenomena. To realize real-time measurement, many works on off-axis dual-wavelength DHM have been reported [21–27]. One way to record interferograms of two wavelengths simultaneously, is by using a color Bayer-mosaic camera, in which two interferograms are separated through different color channels of the camera [21–23]. As two wavelengths are selected to match peak spectral response of Bayer filter of different channels, the synthetic wavelength cannot be too large, as in Ref [21]. it is only 1.9μm, and consequently limited its application for measuring deep steps. Another disadvantage of this way is that the spectral crosstalk between two color channels will lead to additional phase error. The other way to realize simultaneous measurement is with a monochrome camera [24–27], in which two off-axis intergerograms formed by two distinct wavelengths are spatially multiplexed on the sensor. The crossed fringes in the acquired interferogram guarantee the separation of two object spectrum terms in the frequency domain, while each of them belongs to a different wavelength and both of them can be cropped to yield two wrapped phase maps. The main advantage of this method is the synthetic wavelength can be much longer than that in previous method, as selection of two wavelengths is independent of Bayer color filter. However, there are two obvious disadvantages. One is that a camera with large dynamic range is required as two sets of overlapped fringes should be correctly sampled, otherwise quality of the interferogram maybe degraded. The other is that the setup is more complicated since two propagation direction adjustable reference beams must be produced independently. For example, two overlapping Michelson interferometers with its object arm shared are used to construct an apparatus [25].
As a whole, all of these schemes are based on open-path Mach-Zehnder or Michelson interferometers, in which both object and reference beams travel along widely separated paths. The bulky configuration of these interferometers leads to a lower temporal stability, as both beams can be differently affected by environmental disturbances such as mechanical vibrations. The low stability leads to fringe patterns being unstable, which will induce phase error and will become serious in dual-wavelength DHM as being magnified [20], and thus prevents its use in measuring specimens requiring high stability [8]. To solve this problem, common-path interferometry, where object and reference beams travel along the almost the same optical path, is adopted. Although many close to common-path schemes have been proposed to improve the stability, however they are only limited to single-wavelength interferometry [28–35]. Basically, they can be classified into two types according to the generation of reference beam. In Refs [28–31], the reference beam is generated from another copy of object beam with spatially low-filtering technique, in which the information carried by the object beam is erased. The advantage of this method is most kinds of specimens can be imaged, including the case when specimens spread over the field of view. On the contrary, low pass filtering leads to a large portion of energy loss and thus exposure time needs to be longer for compensating low intensity. In addition, a difficult optical alignment is required for pinhole filtering. To overcome these limitations, a self-referencing configuration is presented in Ref [32], in which the reference beam is generated through holographic spatial filtering. However, the Bragg grating used needs to be customized. In Refs [33–35], lateral shearing interferometry or flipping interferometry is adopted, in which a part of object beam containing no specimens is used as reference beam. The main limitation of this method is that it can only image a sample containing a large flat area.
In this paper, we propose a stable dual-wavelength DHM scheme which allowing real-time quantitative phase imaging of deep steps, based on a long working distance objective. The long working distance objective, which has a longer working distance than traditional one, usually used for imaging cells in vitro through thick glass walls. The proposed setup is compact, less complex and easy for alignment while keeping a nearly common path configuration as all the beams travel along almost the same set of optical elements. In the setup, both reference beams of two wavelengths are separated and generated by a broadband polarizing beam splitter, neither through pinhole filtering nor through lateral shearing of object beam, and propagation directions of both beams can be independently adjusted easily by two mirrors. Based on the proposed system, two off-axis interferograms with orthogonal fringe patterns can be simultaneously captured through a monochrome CCD camera.
2. Experimental Setup
The setup for the proposed dual-wavelength off-axis DHM with a long working distance objective is sketched in Fig. 1, which is based on interference multiplexing technique with the help of two orthogonally polarized reference beams. A He-Ne laser (λ1 = 633nm) and a solid state laser (λ2 = 685nm) are used as illumination sources. Two polarizers, P1 and P2, are placed in front of the interferometer, converting the beam of wavelength λ1 into S (vertical) polarization state while converting another beam of λ2 into P (horizontal) polarization state. The two beams are combined by a broadband non-polarizing beam splitter NPBS1, and then, the two collinear beams are expanded by a beam expander BE. An achromatic lens L with focal length of 150 mm and microscope objective MO (NA = 0.28, 10 × ) are placed in a confocal way, i.e., the objective’s pupil plane is coincide with the back focal plane of lens L, to form an inverted telecentric imaging unit, and the advantages of this configuration are illustrated in Ref [13].
Fig. 1 Experimental setup. P1-P2, polarizers; NPBS1-NPBS3, broadband non-polarizing beam splitters; PBS, broadband polarizing beam splitter; BE, beam expander; L, achromatic lens with focal lengths of f = 150 mm; M1-M2, mirrors; MO, 10 × microscope objective with working distance of 33.5 mm; S, sample. Inset, ko is wave vector of both object beams O1 (for λ1) and O2 (for λ2); kr1 and kr2 are wave vectors of the reference beams R1 (for λ1) and R2 (for λ2), respectively;
A small broadband non-polarizing beam splitter NPBS3 is inserted between the MO and the specimen S. After the two beams pass through the MO, half of it illuminating specimen S, while half of it are reflected by NPBS3 to a polarizing beam splitter PBS, for generating two reference beams. The object beams being reflected from the specimen are magnified by MO and are then reshaped by lens L, while being redirected to a monochrome CCD sensor by a non-polarizing beam splitter NPBS2. On the other hand, the PBS separating the beams reflected by NPBS3 into two different wavelengths channels, i.e., the s polarized λ1 beam and the p polarized λ2 beam. Each of the two wavelengths beams is reflected back by mirrors M2 and M1, respectively, to serve as reference beams. And they are also delivered to the CCD sensor at an off-axis angle via MO and lens L. For λ1 wavelength, the reference beam interferences with its corresponding object beam, while for λ2 it forms another interferogram. Hence, two off-axis interferograms of two wavelengths can be simultaneously formed. As the wave vectors of each reference beams can be independently adjusted by tilting mirrors M1 and M2, respectively, the spatial frequencies and orientations of each interferogram can be tuned, such that an off-axis multiplexed interferogram can be obtained, which has orthogonal fringe patterns for each wavelength. The monochrome CCD camera has 1388 × 1040 pixels, 8 bit dynamic range, 6.45 μm × 6.45 μm pixel size, is placed at the focal plane of lens L for recording interferograms.
In the proposed setup, both PBS and NPBS3 are 12.7mm sizes cubes which are placed on a micro stage side by side, while the MO has a working distance of 33.5 mm, hence, there has adequate space for inserting PBS and NPBS3 between MO and object plane. The specimen is placed at a two dimensional translation stage with micrometer positioning precision for searching the object plane, while mirrors M1 and M2 are placed almost attached to PBS to realize a compact design. As shown in Fig. 1, PBS and NPBS3 are combined to construct a miniature modified Michelson interferometer. As the setup is very compact, thus it is less affected by mechanical vibrations and can achieve a higher stable level than previous reported traditional dual-wavelength DHM, as the traditional ones are almost built on interferometer with widely separated object and reference arms [21–27].
3. Reconstruction method
Assume two object beams, O1 for wavelength λ1 and O2 for λ2, travel along the optical axis. While, as seen in the inset of Fig. 1, the reference beam R1 for wavelength λ1 travels in xz plane and has an incidence angle θ1 respect to the wave vector of O1; another reference beam R2 for λ2 travels in yz plane and has an incidence angle θ2 respect to the wave vector of O2. Therefore, the fringe patterns induced by wavelength λ1 across x direction, while the fringe patterns induced by wavelength λ2 across y direction. In the experiment, image-plane holograms are recorded. The intensity distribution, which is an incoherent addition of two interferograms, captured by the CCD camera can be expressed as:
Where, φ1(x, y) represents the phase distribution of the specimen for wavelength λ1; φ2(x, y) represents the phase distribution for wavelength λ2. The first four terms in Eq. (1) represent the zero diffraction order, while the last two terms correspond to the interference terms between the object and reference beams for each wavelength.Figure 2(a) shows one of the captured multiplexed interferograms, and its Fourier spectrum is shown in Fig. 2(b). For such an interferogram, we know that the fringes along x direction has spatial frequency K1 = 2πsinθ1/λ1, and the fringes along y direction has spatial frequency K2 = 2πsinθ2/λ2. These two frequencies can be tuned by adjusting M1 and M2, to guarantee the full separation of two pairs of interference terms (object term and its conjugate) in the Fourier plane. Thus, two object spectrum terms for wavelength λ1, λ2 can be isolated with spatial filtering, separately. The detailed reconstruction procedures for calculating complex object wave Oi(i = 1, 2) are as follows [10]: Firstly, the interferogram I is multiplied with a digital reference beam RDi(x, y) for wavelength λi, in which RD1(x, y) = exp(jK1x), RD2(x,y) = exp(jK2y) given the assumption of unit amplitude, here the spatial frequencies K1 and K2 can be determined from the fringe frequencies of the interferogram I. This operation shifts the spectrum of object beam Oi to the original center in the Fourier plane; Secondly, Fourier transformation of RDI is performed and then the frequency spectrum of object beam Oi can be isolated with a low pass filtering mask Wi, by selecting the central area of the spectrum of RDI; Finally, a Inverse Fourier transformation is performed to get the complex object beam Oi. In summary, the complex object beam for wavelength λi (i = 1, 2) can be calculated with:
Here, FT and IFT represent Fourier transform and Inverse Fourier transform, respectively.Fig. 2 (a) The multiplexed interferogram with orthogonal fringe patterns where inset representing a zoom of the selected area; (b) The Fourier spectrum of the interferogram, in which two object spectrum terms of wavelengths λ1, λ2 located in the two white circles, respectively.
From complex object wave Oi, both wrapped phase maps φi(x, y) can be obtained by taking arctan function, and then phase aberrations presented in φi(x, y) are numerically compensated by least square surface fitting method [36]. However, both of them suffering from phase ambiguities when OPLs are larger than one time of wavelength. In contrary, ambiguity free phase map φ can be calculated with the following expression [23]:
Where, Λ = λ1λ2/|λ1-λ2| is the synthetic wavelength; l denotes OPLs for a specimen, in which coordinates (x, y) are omitted. This synthetic wavelength Λ is much larger than either of the experimental wavelengths λi, the smaller the difference |λ1-λ2| is, the larger the Λ is. Hence, in dual-wavelength DHM, the axial unambiguous measurement range is extended to a synthetic wavelength Λ, ranging from micrometers to millimeters, and this is especially useful when imaging deep steps as in this case it is the only solution in DHM. As the phase noises in synthetic phase φ(x, y) are magnified, to reduce the magnified noises in φ(x, y), firstly each wrapped phase φi(x, y) is denoised by wavelet method, secondly the directly obtained phase φ(x, y) is used as a guiding map to further suppress the noises [20] and finally a fine phase map with noise level achieving that of single wavelength DHM can be obtained. For reflective topography measurement, the height h(x, y) = l(x, y)/2, and hence, the height map h(x, y) of a sample can be calculated with the following formula:4. Experimental results
To demonstrate the capability of the proposed scheme, different reflective specimens with abrupt steps are measured, respectively. In the experiments, two wavelengths are λ1 = 633nm, λ2 = 685nm, respectively, and the synthetic wavelength is Λ = 8.338μm. For our system, we experimentally confirmed the image magnification M = 10.3.
In the first experiment, a part of a standard step sample (VLSI, SHS-1.8QC) is measured. As the nominal height of its pattern is 1.8 μm, which corresponding to 3.6 μm in OPL and is much longer than any of the experimental wavelengths, the sample cannot be measured with single wavelength DHM because the phase unwrapping algorithms cannot tackle the phase ambiguity. With the proposed system, a multiplexed interferogram is captured in one shot and numerically reconstructed. Figure 3(a) shows the wrapped phase map of the object with wavelength λ1 = 633nm, while Fig. 3(b) shows the wrapped phase map with another wavelength λ2 = 685nm. Both of them illustrate phase ambiguities in single wavelength imaging: phase values of bar-like structures are negative in Fig. 3(a) while they are positive as shown in Fig. 3(b). Due to the deep transitions of these structures, phase unwrapping algorithms are helpless. However, after applying dual wavelength measuring procedure, as outlined in the previous section, ambiguity free phase map can be obtained and then correct height map can be calculated according to Eq. (4) as shown in Fig. 3(c). Figure 3(d) shows the histogram of height map, where two dominant peaks appeared at the left and right sides of the figure, one corresponding to the micro-structures and the other corresponding to the base plane. From the difference of two peak values, the step height is calculated to be 1.805μm, which is consistent well with the nominal height. For measuring such a sample with deep steps, it should be noted than even some previously proposed real time dual-wavelength DHM are also useless, as the synthetic wavelength is only 3.33μm, which is less than the OPL of 3.6 μm, due to the reason that in these schemes two wavelengths are selected according to the spectral response of a color camera [22-23].
Fig. 3 Measuring results. (a) Wrapped phase φ1 of the sample obtained with wavelength λ1; (b) Wrapped phase φ2 of the sample obtained with wavelength λ2; (c) Height map H reconstructed from dual wavelength procedure; (d) Histogram of height map in Fig. 3(c). Color bars in Figs. (a-b) represent phase in radians, and color bar in Fig. (c) represents height in μm.
In the second experiment, real time measurement of dynamic process of the proposed setup is demonstrated. The sample to be measured is a reflective micro staircase, which is fabricated on a Si wafer using photolithography. It is comprised of two steps with nominal height values of 200nm and 1000nm, where the maximum OPL corresponding to 2000nm, which is also much longer than either of the experimental wavelengths and out of measurement range of single wavelength DHM. In the experiment, the sample is manually translated along x direction and a sequence of interferograms is captured. Using the described method above, the topographic height map of the micro staircase is retrieved. Figure 4(a) shows one of the height maps, also the real time dynamic process is presented in Visualization 1. The measured average heights of two steps shown in Fig. 4(a) are 209nm and 982nm, respectively, and these slightly deviation from the nominal heights may be due to manufacturing error. To further assess the measurement accuracy, the height profile along the white line in Fig. 4(a) is shown in Fig. 4(b) and compared with the result obtained from a white light scanning interferometer (Talysurf CCI 6000). From Fig. 4(b), we can see that the height measured with the proposed method is in agreement with the result of white light interferometer. Such real time imaging capability is very useful for monitoring moving objects or moving parts of a MEMS structure [24], while classical phase shifting interferometry is helpless.
Fig. 4 Real time measurements of dynamic moving of a micro staircase structure (Visualization 1). (a) One of the reconstructed height map; (b) Height profile obtained with dual wavelength (DW) DHM along the white line in Fig. 4(a) and compared with that measured with a white light interferometer (WLI).
In the final experiments, we measured the temporal stability of the proposed setup and compared the result with that of a traditional dual wavelength interferometer, which is based on Mach-Zehnder configuration with widely separated arms as presented in Ref [24]. A series of interferograms in absence of any specimen, are continuously captured at the rate of 15 frames/s for 20 seconds in each setup under the same conditions. Using the same reconstruction procedure, the phase map is calculated from each frame and then the corresponding height map is obtained. To assess the temporal stability, the standard deviations of 10000 random selected pixels, which locate at the same area of every height map over the sequence, are calculated, respectively. Figure 5(a) shows the histogram of standard deviations distribution of the proposed setup, and its mean value is 2.04 nm. As a counterpart, Fig. 5(b) presents the histogram of standard deviations measured with traditional dual wavelength DHM, indicating a mean value of 10.17 nm. These values are comparable with the results of a shearing common path interferometer as presented in Ref [35]. It should be noted that these values are below that of the state of the art commercial systems (Lyncée Tec), as in our experiments, none of the vibration isolation techniques has been taken and none of the other measures has been adopted to improve the stability. However, from these results, we can conclude that the temporal stability of the proposed setup is improved significantly over the traditional dual wavelength DHM under the same conditions.
Fig. 5 Temporal stability of the proposed setup and the traditional dual wavelength DHM. (a) Histogram of standard deviation of proposed setup; (b) Histogram of standard deviation of dual wavelength DHM based on Mach-Zehnder interferometer.
5. Conclusion
In summary, we have presented a compact dual wavelength DHM scheme based on a long working distance objective, which is capable of measuring samples with deep steps in one-shot. The setup is constructed with simple optics elements and is easy to implement. The compact configuration guarantees a higher temporal stability with mean value of 2.04nm, which means the setup is much stabler than the traditional dual wavelength DHM. In the scheme, two reference beams of each wavelength are generated directly by a PBS. Hence, the multiplexed interferogram can be easily formed by adjusting the propagation directions of two reference beams, respectively. Measurements on both a standard step sample and dynamic process of a staircase illustrated the capability of the system, in which the measurement range is extended to a synthetic wavelength without phase unwrapping. The limitation of the proposed setup is that it only realizable with a relative low N.A. objective, and hence hinders its application in high resolution imaging. Despite this, we still believe that the scheme can be used for topography imaging and metrology, and especially for investigation of fast phenomena for which the single wavelength DHM cannot be resolved.
Acknowledgments
National Natural Science Foundation of China (NSFC) (61605152), Natural Science Basic Research Plan in Shaanxi Province of China (2017JM1037, 2016JQ6053), scientific special research project of Educational Department of Shaanxi Province, China (16JK1359).
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